Search results for " eigenvalues"
showing 10 items of 14 documents
A Teaching proposal for the study of eigenvectors and eigenvalues
2017
[EN] In this work, we present a teaching proposal which emphasizes on visualization and physical applications in the study of eigenvectors and eigenvalues. These concepts are introduced using the notion of the moment of inertia of a rigid body and the GeoGebra software. The proposal was motivated after observing students¿ difficulties when treating eigenvectors and eigenvalues from a geometric point of view. It was designed following a particular sequence of activities with the schema: exploration, introduction of concepts, structuring of knowledge and application, and considering the three worlds of mathematical thinking provided by Tall: embodied, symbolic and formal.
On vibrating thin membranes with mass concentrated near the boundary: an asymptotic analysis
2018
We consider the spectral problem \begin{equation*} \left\{\begin{array}{ll} -\Delta u_{\varepsilon}=\lambda(\varepsilon)\rho_{\varepsilon}u_{\varepsilon} & {\rm in}\ \Omega\\ \frac{\partial u_{\varepsilon}}{\partial\nu}=0 & {\rm on}\ \partial\Omega \end{array}\right. \end{equation*} in a smooth bounded domain $\Omega$ of $\mathbb R^2$. The factor $\rho_{\varepsilon}$ which appears in the first equation plays the role of a mass density and it is equal to a constant of order $\varepsilon^{-1}$ in an $\varepsilon$-neighborhood of the boundary and to a constant of order $\varepsilon$ in the rest of $\Omega$. We study the asymptotic behavior of the eigenvalues $\lambda(\varepsilon)$ and the eige…
Elementary presentation of self‐consistent intermediate Hamiltonians and proposal of two totally dressed singles and doubles configuration interactio…
1994
Intermediate Hamiltonians are effective Hamiltonians which are defined on an N‐dimensional model space but which only provide n<N exact eigenvalues and the projections of the corresponding eigenvectors onto the model space. For a single root research, the intermediate Hamiltonian may be obtained from the restriction of the Hamiltonian to the model space by an appropriate, uniquely defined dressing of the diagonal energies or of the first column. Approximate self‐consistent dressings may be proposed. The simplest perturbative form gives the same result as the original 2nd order intermediate Hamiltonian or the ‘‘shifted Bk’’ technique but it is of easier implementation. Self‐consistent inclus…
The equality case in a Poincaré–Wirtinger type inequality
2016
It is known that, for any convex planar set W, the first non-trivial Neumann eigenvalue μ1 (Ω) of the Hermite operator is greater than or equal to 1. Under the additional assumption that Ω is contained in a strip, we show that β1 (Ω) = 1 if and only if Ω is any strip. The study of the equality case requires, among other things, an asymptotic analysis of the eigenvalues of the Hermite operator in thin domains.
Sharp estimates for eigenfunctions of a Neumann problem
2009
In this paper we provide some bounds for the eigenfunctions of the Laplacian with homogeneous Neumann boundary conditions in a bounded domain Ω of R^n. To this aim we use the so-called symmetrization techniques and the obtained estimates are asymptotically sharp, at least in the bidimensional case, when the isoperimetric constant relative to Ω goes to 0.
Analysis of the finite difference time domain technique to solve the Schrödinger equation for quantum devices
2004
An extension of the finite difference time domain is applied to solve the Schrödinger equation. A systematic analysis of stability and convergence of this technique is carried out in this article. The numerical scheme used to solve the Schrödinger equation differs from the scheme found in electromagnetics. Also, the unit cell employed to model quantum devices is different from the Yee cell used by the electrical engineering community. A bound for the time step is derived to ensure stability. Several numerical experiments in quantum structures demonstrate the accuracy of a second order, comparable to the analysis of electromagnetic devices with the Yee cell. a!Electronic mail: Antonio.Sorian…
Eigenvalues of non-hermitian matrices: a dynamical and an iterative approach. Application to a truncated Swanson model
2020
We propose two different strategies to find eigenvalues and eigenvectors of a given, not necessarily Hermitian, matrix (Formula presented.). Our methods apply also to the case of complex eigenvalues, making the strategies interesting for applications to physics and to pseudo-Hermitian quantum mechanics in particular. We first consider a dynamical approach, based on a pair of ordinary differential equations defined in terms of the matrix (Formula presented.) and of its adjoint (Formula presented.). Then, we consider an extension of the so-called power method, for which we prove a fixed point theorem for (Formula presented.) useful in the determination of the eigenvalues of (Formula presented…
The second Weyl coefficient for a first-order system
2020
For a scalar elliptic self-adjoint operator on a compact manifold without boundary we have two-term asymptotics for the number of eigenvalues between 0 and λ when λ → ∞, under an additional dynamical condition. (See [3, Theorem 3.5] for an early result in this direction.) In the case of an elliptic system of first order, the existence of two-term asymptotics was also established quite early and as in the scalar case Fourier integral operators have been the crucial tool. The complete computation of the coefficient of the second term was obtained only in the 2013 paper [2]. In the present paper we simplify that calculation. The main observation is that with the existence of two-term asymptoti…
The affine equivariant sign covariance matrix: asymptotic behavior and efficiencies
2003
We consider the affine equivariant sign covariance matrix (SCM) introduced by Visuri et al. (J. Statist. Plann. Inference 91 (2000) 557). The population SCM is shown to be proportional to the inverse of the regular covariance matrix. The eigenvectors and standardized eigenvalues of the covariance, matrix can thus be derived from the SCM. We also construct an estimate of the covariance and correlation matrix based on the SCM. The influence functions and limiting distributions of the SCM and its eigenvectors and eigenvalues are found. Limiting efficiencies are given in multivariate normal and t-distribution cases. The estimates are highly efficient in the multivariate normal case and perform …
Estimates for Sums of Eigenvalues of the Free Plate via the Fourier Transform
2017
Using the Fourier transform, we obtain upper bounds for sums of eigenvalues of the free plate.